Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $y = \dfrac{3n}{6(3n + 1)} \div \dfrac{3n}{10(3n + 1)} $
Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{3n}{6(3n + 1)} \times \dfrac{10(3n + 1)}{3n} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 3n \times 10(3n + 1) } { 6(3n + 1) \times 3n } $ $ y = \dfrac{30n(3n + 1)}{18n(3n + 1)} $ We can cancel the $3n + 1$ so long as $3n + 1 \neq 0$ Therefore $n \neq -\dfrac{1}{3}$ $y = \dfrac{30n \cancel{(3n + 1})}{18n \cancel{(3n + 1)}} = \dfrac{30n}{18n} = \dfrac{5}{3} $